parallel curve

Given two curves, one is a parallel curve (also known as an offset curve) of the other if the points on the first curve are equidistant to the corresponding points in the direction of the second curve’s normal. Alternatively, a parallel of a curve can be defined as the envelope of congruent circles whose centers lie on the curve.

For a parametric curve in the plane defined by $\vec{F}(u):=(x(u),\,y(u))$ , its parallel curve $\vec{G}(u):=(X(u),\,Y(u))$ with offset $t$ is defined by

	$\displaystyle X(u)$	$\displaystyle=\,x(u)\!+\!\frac{t\,y^{\prime}(u)}{\sqrt{x^{\prime}(u)^{2}\!+y^{% \prime}(u)^{2}}}$
	$\displaystyle Y(u)$	$\displaystyle=\,y(u)\!-\!\frac{t\,x^{\prime}(u)}{\sqrt{x^{\prime}(u)^{2}\!+y^{% \prime}(u)^{2}}}$

0.1 Examples

The most elementary example of parallel curves is given by the family of concentric circles

	$\displaystyle X(u)$	$\displaystyle=$	$\displaystyle t\cos u$
	$\displaystyle Y(u)$	$\displaystyle=$	$\displaystyle t\sin u$

Except for trivial cases such as circles and lines, parallel curves may be quite different from the original curve as the offset gets larger. An example of this is given by the catenary

	$\displaystyle x(u)$	$\displaystyle=$	$\displaystyle u$
	$\displaystyle y(u)$	$\displaystyle=$	$\displaystyle\cosh{u}$

From the definition, the family of parallel curves is then

	$\displaystyle X$	$\displaystyle=$	$\displaystyle u+\frac{t\sinh{u}}{\sqrt{1+\sinh^{2}{u}}}\,=\,u+t\tanh{u}$
	$\displaystyle Y$	$\displaystyle=$	$\displaystyle\cosh{u}-\frac{t}{\sqrt{1+\sinh^{2}{u}}}\,=\,\cosh{u}-\frac{t}{% \cosh{u}}$

where $t=0$ correspond to the catenary.

Eliminating the parameter $u$ from these equations; the latter gives $\cosh{u}=\frac{Y+\sqrt{Y^{2}+4t}}{2}$ , i.e. $u=\operatorname{arcosh}\frac{Y+\sqrt{Y^{2}+4t}}{2}$ . Thus we obtain the implicit representation

\operatorname{arcosh}\frac{Y\!+\!\sqrt{Y^{2}\!+\!4t}}{2}+t\,\tanh\!\left(\!% \operatorname{arcosh}\frac{Y\!+\!\sqrt{Y^{2}\!+\!4t}}{2}\right)-X\,=\,0

Title	parallel curve
Canonical name	ParallelCurve
Date of creation	2013-03-22 17:13:10
Last modified on	2013-03-22 17:13:10
Owner	stitch (17269)
Last modified by	stitch (17269)
Numerical id	21
Author	stitch (17269)
Entry type	Definition
Classification	msc 51N05
Synonym	offset curve
Related topic	ParallellismInEuclideanPlane
Related topic	NormalLine
Related topic	HyperbolicFunctions